Lecturer: Matteo Negri   Giuseppe Savarè  

Course name: Mathematical Methods for Engineers
Course code: 502461
Degree course: Bioingegneria
Disciplinary field of science: MAT/05
L'insegnamento costituisce attività di base per: Bioingegneria
University credits: CFU 6
Course website: http://www.imati.cnr.it/~savare/didattica/metodi/

Specific course objectives

Students will be introduced to the basic mathematical tools for signal theory and optimization. To this aim, the course is divided in two parts.
In the first part, MATHEMATICAL METHODS (6CFU), they will learn how to work in the complex framework, evaluate integrals of olomorphic functions, manipulate power and Fourier series, adopt the point of view of signal theory, calculate and operate with Z, Fourier and Laplace transforms, solve simple ordinary differential equations with constant coefficients, understand convolutions.
The second part, OPTIMIZATION AND DISCRETE TRANSFORMS (3CFU), will be devoted to the elementary notions of free and constraint optimization and to the basic techniques of the mathematical theory of discrete signals (DFT, FFT, convolutions) with simple applications to difference equations and numerical approximations.

Course programme

Complex functions

• Manipulation of complex numbers
• Rational, exponential, and trigonometric functions, logarithms
• Power series
• Conplex derivatives, olomorphic functions, Cauchy-Riemann conditions
• Line integrals, Cauchy theorem, , analyticity of olomorphic functions
• Singularities, Laurent series, residue formula
• Evaluation of integrals, Jordan lemma

Signals

• Discrete and continuous signals
• Elementary manipulation of signals: sum, linear combination, shift and rescaling.
• Scalar products and norms

Z transform

• Definition, simple properties, examples
• Applications to linear difference equations

Fourier series

• Periodic signals, trigonometric and exponential functions, Fourier series.
• Pointwise and energy convergence, Gibbs phenomenon.
• Parseval identity
• Applications

Fourier trasform

• Definition of Fourier transform, relationships with Fourier series, elementary properties
• Riemann-Lebesgue lemma
• Inversion theorem for piecewise regular functions
• Plancherel identity, Fourier transform for L^2 functions

Laplace transform

• Definition, links with the Fourier transform, main properties
• Inversion of Laplace transform, residue and Heaviside formula
• Application to simple ordinary differential equations

Convolutions

• Definition and simple example of convolutions
• Links with Fourier and Laplace transform
• Simple applications to differential equations

Optimization

• Unconstrained Optimization Problems
  - Gradient methods and line-searches
  - Newtonian methods: trust-regions, quasi-Newton and Gauss-Newton for least-squares problems
• Constrained Optimization Problems
  - Optimality conditions
  - Newton's method for problems with equality constraints
  - SQP method
  - penalization/barrier methods

Discrete transforms

• Discrete Fourier transform (DFT)
• The algorithm of Fast Fourier Transform (FFT)
• Discrete convolution
• Applications to difference and approximation problems, stability

Course entry requirements

Differential and integral calculus for scalar and vector functions, matrices and linear transformations, sequences and series, power series in the real line, complex numbers, polar coordinates.

Course structure and teaching

Lectures (hours/year in lecture theatre): 37
Practical class (hours/year in lecture theatre): 58
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

M. Codegone. Metodi matematici per l'Ingegneria. Zanichelli.

M. Giaquinta, G. Modica. Note di Metodi Matematici per Ingegneria Informatica. Pitagora, Bologna.

F. Tomarelli. Esercizi di Metodi Matematici per l'Ingegneria. CLU.

G. Savaré. Lecture notes. The pdf file can be downloaded from the web site of the course.

Matlab Optimization and Signal Proccessing Toolbox. User's guide. The MathWorks Inc..

F.J. Bonnan, C.J. Gilbert, C. Lemarechal C, C.A. Sagastizabal. Numerical Optimization. Theoretical and practical aspects. Springer Verlag (Universitext), 2006. Second edition.

Testing and exams

Written and oral examination